Algebraic Solutions of Differential Equations
95
To conclude, we simply observe that as
car
cbr
the
inequalities in (6.6.0.15) and (6.6.0.16), already established, are neces
sarily
strict.
This concludes the proof of (6.6.0).
(6.6.1)
Combining 6.3 and 6.6.0, we are "reduced" to proving the
implication (6.2.6)~ (6.2.4) under the additional hypothesis that the
rational numbers a, b, c satisfy
(6.6.1.0)
aCZ, b6Z, c~Z,
caCZ,
cbc~Z, ab6Z,
cab~Z.
(6.6.2)
Corollary.
In order that rational numbers a, b, c satisfy
(6.2.6)
and
(6.6.1.0),
it is necessary and sufficient that for any A eZ which is invertible
modulo N for a common denominator N of a, b, c, the rational numbers
A a, A b, A c satisfy
(6.2.6)
and
(6.6.1.0).
Proof
The sufficiency is clear; take A = 1. For the necessity, (6.6.1.0)
will still hold because d is invertible mod N, and the condition (6.6.0.2)
is obviously invariant under (a, b, c)* (A a, A b, A c) for such A. By
(6.6.0), (6.6.0.2) and (6.6.1.0) imply (6.2.6).
(6.6.3)
Corollary.
In order that rational numbers a, b, c satisfy
(6.2.6)
and
(6.6.1.0),
it is necessary and sufficient that for any integers r, s, t, the
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 Fall '11
 NormanKatz
 Rational number

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